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On solvability of the Cauchy problem for one quasilinear singular functional-differential equation

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Abstract

We consider the Cauchy problem with zero initial conditions for quasilinear singular functional-differential equation of the second order with a delay at singular summand. We obtain sufficient conditions of solvability of the problem.

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References

  1. Shragin, I. V. “On the Carathéodory Conditions”, Russ. Math. Surv. 34, No. 3, 220–221 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  2. Kiguradze, I. T. and Shekhter, B. D. “Singular Boundary-Value Problems for Second-Order Ordinary Differential Equations”, J. SovietMath. 43, 2340–2417 (1988).

    Google Scholar 

  3. Azbelev, N. V., Maksimov, V. P. and Rakhmatullina, L. F. Elements of Modern Theory of Functional-Differential Equations. Methods and Applications (Institute of Computer Investigations, Moscow, 2002) [in Russian].

    Google Scholar 

  4. Fadeev, L. D. and Yakubovskii, L. D. Lectures on Quantum Mechanics for Students-Mathematicians (Leningrad Univ. Press, Leningrad, 1980) [in Russian].

    Google Scholar 

  5. Simonov, N. I. Applied Analysis Methods of Euler (GITTL, Moscow, 1957) [in Russian].

    Google Scholar 

  6. Norkin, S. B. “Structure of Solutions to Differential Equation with Delayed Argument in a Neighborhood of a Singular Point”, Pr. a štud. VS dopravy a spojov Ziline. Sér. mat. -fyz., No. 3, 27–46 (1980).

    MathSciNet  MATH  Google Scholar 

  7. Fišnarová, S. and Mařík, R. “Oscillation Criteria for Neutral Second-order Half-linear Differential Equations with Applications to Euler Type Equations”, Boundary Value Problems, 2014, 2014:83 (www. boundaryvalueproblems. com/content/2014/1/83).

    Google Scholar 

  8. Kiguradze, I. and Sokhadze, Z. “On the Global Solvability of the Cauchy Problem for Singular Functional Differential Equations”, GeorgianMath. Journal 4, No. 4, 355–372 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  9. Labovskii, S. M. “Positive Solutions of a Two-Point Boundary Value Problem for a Linear Singular Functional-differential Equation”, Differential Equations 24 (10), 1116–1123 (1988).

    MathSciNet  Google Scholar 

  10. Shindyapin, A. I. “A Boundary Value Problem for a Singular Equation”, Differentsial’nye Uravneniya 20 (3), 450–455 (1984).

    MathSciNet  MATH  Google Scholar 

  11. Alvesh, M. Zh. “A Nonlinear Boundary Value Problem with a Non-Summable Singularity”, Russian Mathematics (Iz. VUZ) 44, No. 4, 54–57 (2000).

    MathSciNet  Google Scholar 

  12. Bravyi, E. I. “On the Solvability of a Two-Point Boundary Value Problem for a Linear Singular Functional Differential Equation”, RussianMathematics (Iz. VUZ) 48, No. 6, 12–18 (2004).

    MathSciNet  MATH  Google Scholar 

  13. Plaksina, I. M. “On Positiveness of the Cauchy Function of a Singular Linear Functional Differential Equation”, RussianMathematics (Iz. VUZ) 57, No. 10, 13–18 (2013).

    MathSciNet  MATH  Google Scholar 

  14. Muntean, I. “The Spectrum of the Cesaro Operator”, Mathematica. Mathématica. Revue d’analyse numérique et de théorie de l’approximation 22 (45), 97–105 (1980).

    MathSciNet  Google Scholar 

  15. Bragina, N. A. “Solvability of Boundary Value Problems for Quasilinear Functional-Differential Equations”, Candidate’s Dissertation in Mathematics and Physics (Perm’, 2004).

    Google Scholar 

  16. Abdullaev, A. R. and Plaksina, I. M. “An Estimate of the Spectral Radius of a Certain Singular Integral Operator”, RussianMathematics (Iz. VUZ) 59, No. 2, 1–6 (2015).

    MathSciNet  Google Scholar 

  17. Abdullaev, A. R. and Plekhova, E. V. “About Spectrum of Cesàro Operators”, Nauchno-Tekhn. Vestnik Povolzh’ya, No. 4, 33–37 (2011).

    Google Scholar 

  18. Halmos, P. R. and Sunder, V. S. Bounded Integral Operators on L2 Spaces (Springer-Verlag, 1978; Nauka, Moscow 1985).

    Google Scholar 

  19. Drakhlin, M. E. and Plyshevskaya, T. K. “On the Theory of Functional-Differential Equations”, Differentsial’nye Uravneniya 14, No. 8, 1347–1361 (1978).

    MathSciNet  MATH  Google Scholar 

  20. Riesz, F. and Nagy, B. Sz. Leçons d’Analyse Fonctionnelle (Gauthier-Villars, Paris, 1975; Mir, Moscow, 1979).

    Google Scholar 

  21. Maksimov, V. P. “Some Nonlinear Boundary Value Problems”, Differentsial’nyeUravneniya 19 (3), 396–414 (1983).

    MathSciNet  Google Scholar 

  22. Trenogin, V. A. Functional Analysis (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

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Authors and Affiliations

  1. State National Research Polytechnical University of Perm, Komsomolskii pr. 29, Perm, 614990, Russia

    V. P. Plaksina, I. M. Plaksina & E. V. Plekhova

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  1. V. P. Plaksina
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  2. I. M. Plaksina
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  3. E. V. Plekhova
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Correspondence to V. P. Plaksina.

Additional information

Original Russian Text © V.P. Plaksina, I.M. Plaksina, E.V. Plekhova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 2, pp. 54–61.

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Plaksina, V.P., Plaksina, I.M. & Plekhova, E.V. On solvability of the Cauchy problem for one quasilinear singular functional-differential equation. Russ Math. 60, 46–51 (2016). https://doi.org/10.3103/S1066369X16020080

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